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Proof of Karlin-Rubin's theorem, detail about a real analysis fact.
Although the setting of this question is statistics, the question actually asks for a real analysis fact (monotone functions). Karlin-Rubin's theorem...
Apr 15, 2026
Archive: News
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Partial Fraction Expansion of Transfer Function
How do I go from: $$ \frac{3(1+0.2z^{-1})(1+z^{-1})}{(1+0.5z^{-1})(1-0.4z^{-1})} $$ to $$ -3 + \frac{7}{1-0.4z^{-1}} - \frac{1}{1+0.5z^{-1}} $$ I understa...
Apr 15, 2026
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Why angle form by a secant and tangent outside a circle is half the intercepted arc?
I know that according to Inscribed Angle Theorem that the measure of an angle will be half the intercepted arc's measure. I know that the measure of ...
Apr 15, 2026
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How to find the area of the following isosceles triangle
I am stuck with the following problem : What is the area of an isosceles triangle whose equal sides are $20$ cm and the angle between them is $30^{\circ}$...
Apr 15, 2026
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Mentally calculating trigonometric function values such as $\sin(47^\circ)$
This may sound dumb, but does such a way exist to mentally (and quickly) determine the values of trigonometric functions such as $\sin(47^\circ)$ and so f...
Apr 15, 2026
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Factor $3x^{3/2}-9x^{1/2}+6x^{-1/2}$
I have $3x^{-1/2} (x^2-3x+2)$ However, I just tried to expand, and the answer is not the same as the original question. With fractional exponents I take o...
Apr 15, 2026
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How many solutions are there to $abc+def=ghi$, where $a,b,\ldots, h,i$ are distinct non-zero digits?
I saw this problem posted by Google. Those posting in the comments found solutions using computer programming. I would like to know if there is an easier ...
Apr 15, 2026
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Does the square or the circle have the greater perimeter? A surprisingly hard problem for high schoolers
An exam for high school students had the following problem: Let the point $E$ be the midpoint of the line segment $AD$ on the square $ABCD$. Then let a ci...
Apr 15, 2026
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What is a paradox in mathematics?
I was reading some of these "mathematical paradoxes", and trying to understand why the list presents only counterintuitive mathematical results. Is there ...
Apr 15, 2026
![Find value of $x^{5} + y^{5} + z^{5}$ given the values of $x + y + z$, $x^{2} + y^{2} + z^{2}$ and $x^{3} + y^{3} + z^{3}$ [closed]](https://news.conceivablytech.com/uploads/find-value-of-x-5-y-5-z-5-given-the-values-of-x-y-z-x-2-y-2-z-2-and-x-3-y-3-z-3-closed_720.jpg)
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Find value of $x^{5} + y^{5} + z^{5}$ given the values of $x + y + z$, $x^{2} + y^{2} + z^{2}$ and $x^{3} + y^{3} + z^{3}$ [closed]
If$$x+y+z=1$$ $$x^2+y^2+z^2=2$$ $$x^3+y^3+z^3=3$$ Then find the value of $$x^5+y^5+z^5$$ Is there any simple way to solve this problem ? I have tried all ...
Apr 15, 2026
